Using the psych package to generate and test structural models

Using the psych package to generate and test structural models William Revelle July 7, 2015 Contents 1 The psych package 3 1.1 Preface...................................... 3 1.2 Creating and modeling structural relations................... 3 2 Functions for generating correlational matrices with a particular structure 4 2.1 sim.congeneric................................. 5 2.2 sim.hierarchical................................ 7 2.3 sim.item and sim.circ............................. 8 2.4 sim.structure.................................. 8 2.4.1 f x is a vector implies a congeneric model............... 8 2.4.2 f x is a matrix implies an independent factors model:......... 11 2.4.3 f x is a matrix and Phi I is a correlated factors model....... 11 2.4.4 f x and f y are matrices, and Phi I represents their correlations.. 15 2.4.5 A hierarchical structure among the latent predictors.......... 16 3 Exploratory functions for analyzing structure 16 3.1 Exploratory simple structure models...................... 19 3.2 Exploratory hierarchical models......................... 23 3.2.1 A bifactor solution............................ 23 3.2.2 A hierarchical solution.......................... 23 4 Confirmatory models 28 4.1 Using psych as a front end for the sem package................ 28 4.2 Testing a congeneric model versus a tau equivalent model.......... 28 4.3 Testing the dimensionality of a hierarchical data set by creating the model. 30 4.4 Testing the dimensionality based upon an exploratory analysis....... 32 1

4.5 Specifying a three factor model......................... 32 4.6 Allowing for an oblique solution......................... 33 4.7 Extract a bifactor solution using omega and then test that model using sem 35 4.7.1 sem of Thurstone 9 variable problem.................. 35 4.8 Examining a hierarchical solution........................ 38 4.9 Estimating Omega using EFA followed by CFA................ 42 5 Summary and conclusion 43 2

1 The psych package 1.1 Preface The psych package (Revelle, 2014) has been developed to include those functions most useful for teaching and learning basic psychometrics and personality theory. Functions have been developed for many parts of the analysis of test data, including basic descriptive statistics (describe and pairs.panels), dimensionality analysis (ICLUST, VSS, principal, factor.pa), reliability analysis (omega, guttman) and eventual scale construction (cluster.cor, score.items). The use of these and other functions is described in more detail in the accompanying vignette (overview.pdf) as well as in the complete user s manual and the relevant help pages. (These vignettes are also available at http:// personality-project.org/r/overview.pdf) and http://personality-project.org/ r/psych_for_sem.pdf). This vignette is concerned with the problem of modeling structural data and using the psych package as a front end for the much more powerful sem package of John Fox (Fox, 2006, 2009; Fox et al., 2013). Future releases of this vignette will include examples for using the lavaan package of Yves Rosseel (Rosseel, 2012). The first section discusses how to simulate particular latent variable structures. The second considers several Exploratory Factor Analysis (EFA) solutions to these problems. The third section considers how to do confirmatory factor analysis and structural equation modeling using the sem package but with the input prepared using functions in the psych package. 1.2 Creating and modeling structural relations One common application of psych is the creation of simulated data matrices with particular structures to use as examples for principal components analysis, factor analysis, cluster analysis, and structural equation modeling. This vignette describes some of the functions used for creating, analyzing, and displaying such data sets. The examples use two other packages: Rgraphviz and sem. Although not required to use the psych package, sem is required for these examples. Although Rgraphviz had been used for the graphical displays, it has now been replaced with graphical functions within psych. The analyses themselves require only the sem package to do the structural modeling. 3

2 Functions for generating correlational matrices with a particular structure The sim family of functions create data sets with particular structure. Most of these functions have default values that will produce useful examples. Although graphical summaries of these structures will be shown here, some of the options of the graphical displays will be discussed in a later section. The sim functions include: sim.structure A function to combine a measurement and structural model into one data matrix. Useful for understanding structural equation models. Combined with structure.diagram to see the proposed structure. sim.congeneric A function to create congeneric items/tests for demonstrating classical test theory. This is just a special case of sim.structure. sim.hierarchical A function to create data with a hierarchical (bifactor) structure. sim.general A function to simulate a general factor and multiple group factors. This is done in a somewhat more obvious, although less general, method than sim.hierarcical. sim.item A function to create items that either have a simple structure or a circumplex structure. sim.circ Create data with a circumplex structure. sim.dichot Create dichotomous item data with a simple or circumplex structure. sim.minor Create a factor structure for nvar variables defined by nfact major factors and minor factors for n observations. nvar 2 sim.parallel Create a number of simulated data sets using sim.minor to show how parallel analysis works. sim.rasch Create IRT data following a Rasch model. sim.irt Create a two parameter IRT logistic (2PL) model. sim.anova Simulate a 3 way balanced ANOVA or linear model, with or without repeated measures. Useful for teaching courses in research methods. To make these examples replicable for readers, all simulations are prefaced by setting the random seed to a fixed (and for some, memorable) number (Adams, 1980). For normal use of the simulations, this is not necessary. 4

2.1 sim.congeneric Classical test theory considers tests to be tau equivalent if they have the same covariance with a vector of latent true scores, but perhaps different error variances. Tests are considered congeneric if they each have the same true score component (perhaps to a different degree) and independent error components. The sim.congeneric function may be used to generate either structure. The first example considers four tests with equal loadings on a latent factor (that is, a τ equivalent model). If the number of subjects is not specified, a population correlation matrix will be generated. If N is specified, then the sample correlation matrix is returned. If the short option is FALSE, then the population matrix, sample matrix, and sample data are all returned as elements of a list. > library(psych) > set.seed(42) > tau <- sim.congeneric(loads=c(.8,.8,.8,.8)) #population values > tau.samp <- sim.congeneric(loads=c(.8,.8,.8,.8),n=100) # sample correlation matrix for 100 cases > round(tau.samp,2) V1 V2 V3 V4 V1 1.00 0.68 0.72 0.66 V2 0.68 1.00 0.65 0.67 V3 0.72 0.65 1.00 0.76 V4 0.66 0.67 0.76 1.00 > tau.samp <- sim.congeneric(loads=c(.8,.8,.8,.8),n=100, short=false) > tau.samp Call: NULL $model (Population correlation matrix) V1 V2 V3 V4 V1 1.00 0.64 0.64 0.64 V2 0.64 1.00 0.64 0.64 V3 0.64 0.64 1.00 0.64 V4 0.64 0.64 0.64 1.00 $r (Sample correlation matrix for sample size = 100 ) V1 V2 V3 V4 V1 1.00 0.70 0.62 0.58 V2 0.70 1.00 0.65 0.64 V3 0.62 0.65 1.00 0.59 V4 0.58 0.64 0.59 1.00 > dim(tau.samp$observed) [1] 100 4 In this last case, the generated data are retrieved from tau.samp$observed. Congeneric data are created by specifying unequal loading values. The default values are loadings of c(.8,.7,.6,.5). As seen in Figure 1, tau equivalence is the special case where all paths are equal. 5

> cong <- sim.congeneric(n=100) > round(cong,2) V1 V2 V3 V4 V1 1.00 0.57 0.53 0.46 V2 0.57 1.00 0.35 0.41 V3 0.53 0.35 1.00 0.43 V4 0.46 0.41 0.43 1.00 > #plot.new() > m1 <- structure.diagram(c("a","b","c","d")) Structural model x1 x2 x3 a b c d X1 x4 Figure 1: Tau equivalent tests are special cases of congeneric tests. Tau equivalence assumes a=b=c=d 6

2.2 sim.hierarchical The previous function, sim.congeneric, is used when one factor accounts for the pattern of correlations. A slightly more complicated model is when one broad factor and several narrower factors are observed. An example of this structure might be the structure of mental abilities, where there is a broad factor of general ability and several narrower factors (e.g., spatial ability, verbal ability, working memory capacity). Another example is in the measure of psychopathology where a broad general factor of neuroticism is seen along with more specific anxiety, depression, and aggression factors. This kind of structure may be simulated with sim.hierarchical specifying the loadings of each sub factor on a general factor (the g-loadings) as well as the loadings of individual items on the lower order factors (the f-loadings). An early paper describing a bifactor structure was by Holzinger and Swineford (1937). A helpful description of what makes a good general factor is that of Jensen and Weng (1994). For those who prefer real data to simulated data, six data sets are included in the bifactor data set. One is the original 14 variable problem of Holzinger and Swineford (1937) (holzinger), a second is a nine variable problem adapted by Bechtoldt (1961) from Thurstone and Thurstone (1941) (the data set is used as an example in the SAS manual and discussed in great detail by McDonald (1999)), a third is from a recent paper by Reise et al. (2007) with 16 measures of patient reports of interactions with their health care provider. > set.seed(42) > gload=matrix(c(.9,.8,.7),nrow=3) > fload <- matrix(c(.8,.7,.6,rep(0,9),.7,.6,.5, + rep(0,9),.7,.6,.4), ncol=3) > fload #echo it to see the structuresw [,1] [,2] [,3] [1,] 0.8 0.0 0.0 [2,] 0.7 0.0 0.0 [3,] 0.6 0.0 0.0 [4,] 0.0 0.7 0.0 [5,] 0.0 0.6 0.0 [6,] 0.0 0.5 0.0 [7,] 0.0 0.0 0.7 [8,] 0.0 0.0 0.6 [9,] 0.0 0.0 0.4 > bifact <- sim.hierarchical(gload=gload,fload=fload) > round(bifact,2) V1 V2 V3 V4 V5 V6 V7 V8 V9 V1 1.00 0.56 0.48 0.40 0.35 0.29 0.35 0.30 0.20 V2 0.56 1.00 0.42 0.35 0.30 0.25 0.31 0.26 0.18 V3 0.48 0.42 1.00 0.30 0.26 0.22 0.26 0.23 0.15 V4 0.40 0.35 0.30 1.00 0.42 0.35 0.27 0.24 0.16 V5 0.35 0.30 0.26 0.42 1.00 0.30 0.24 0.20 0.13 V6 0.29 0.25 0.22 0.35 0.30 1.00 0.20 0.17 0.11 V7 0.35 0.31 0.26 0.27 0.24 0.20 1.00 0.42 0.28 7

V8 0.30 0.26 0.23 0.24 0.20 0.17 0.42 1.00 0.24 V9 0.20 0.18 0.15 0.16 0.13 0.11 0.28 0.24 1.00 These data can be represented as either a bifactor (Figure 2 panel A) or hierarchical (Figure 2 Panel B) factor solution. The analysis was done with the omega function. 2.3 sim.item and sim.circ Many personality questionnaires are thought to represent multiple, independent factors. A particularly interesting case is when there are two factors and the items either have simple structure or circumplex structure. Examples of such items with a circumplex structure are measures of emotion (Rafaeli and Revelle, 2006) where many different emotion terms can be arranged in a two dimensional space, but where there is no obvious clustering of items. Typical personality scales are constructed to have simple structure, where items load on one and only one factor. An additional challenge to measurement with emotion or personality items is that the items can be highly skewed and are assessed with a small number of discrete categories (do not agree, somewhat agree, strongly agree). The more general sim.item function, and the more specific, sim.circ functions simulate items with a two dimensional structure, with or without skew, and varying the number of categories for the items. An example of a circumplex structure is shown in Figure 3 2.4 sim.structure A more general case is to consider three matrices, f x, φ xy, f y which describe, in turn, a measurement model of x variables, f x, a measurement model of y variables, f x, and a covariance matrix between and within the two sets of factors. If f x is a vector and f y and phi xy are NULL, then this is just the congeneric model. If f x is a matrix of loadings with n rows and c columns, then this is a measurement model for n variables across c factors. If phi xy is not null, but f y is NULL, then the factors in f x are correlated. Finally, if all three matrices are not NULL, then the data show the standard linear structural relations (LISREL) structure. Consider the following examples: 2.4.1 f x is a vector implies a congeneric model > set.seed(42) > fx <- c(.9,.8,.7,.6) 8

> op <- par(mfrow=c(1,2)) > m.bi <- omega(bifact,title="a bifactor model") > m.hi <- omega(bifact,sl=false,title="a hierarchical model") > op <- par(mfrow = c(1,1)) A bifactor model A hierarchical model V1 V1 0.7 0.6 V2 V3 0.3 0.3 0.3 F1* V2 V3 0.8 0.7 0.6 F1 0.5 0.6 V4 0.4 V4 0.7 0.9 g 0.5 V5 0.4 F2* V5 0.6 F2 0.8 g 0.4 0.5 V6 0.3 V6 0.5 0.7 0.4 0.3 V7 V8 0.5 0.4 0.3 F3* V7 V8 0.7 0.6 0.4 F3 V9 V9 Figure 2: (Left panel) A bifactor solution represents each test in terms of a general factor and a residualized group factor. (Right Panel) A hierarchical factor solution has g as a second order factor accounting for the correlations between the first order factors 9

> circ <- sim.circ(16) > f2 <- fa(circ,2) > plot(f2,title="16 simulated variables in a circumplex pattern") 16 simulated variables in a circumplex pattern MR2 0.6 0.4 0.2 0.0 0.2 0.4 0.6 8 9 7 6 10 5 4 11 12 3 13 2 14 1 15 16 0.6 0.4 0.2 0.0 0.2 0.4 0.6 MR1 Figure 3: Emotion items or interpersonal items frequently show a circumplex structure. Data generated by sim.circ and factor loadings found by the principal axis algorithm using factor.pa. 10

> cong1 <- sim.structure(fx) > cong1 Call: sim.structure(fx = fx) $model (Population correlation matrix) V1 V2 V3 V4 V1 1.00 0.72 0.63 0.54 V2 0.72 1.00 0.56 0.48 V3 0.63 0.56 1.00 0.42 V4 0.54 0.48 0.42 1.00 $reliability (population reliability) [1] 0.81 0.64 0.49 0.36 2.4.2 f x is a matrix implies an independent factors model: > set.seed(42) > fx <- matrix(c(.9,.8,.7,rep(0,9),.7,.6,.5,rep(0,9),.6,.5,.4), ncol=3) > three.fact <- sim.structure(fx) > three.fact Call: sim.structure(fx = fx) $model (Population correlation matrix) V1 V2 V3 V4 V5 V6 V7 V8 V9 V1 1.00 0.72 0.63 0.00 0.00 0.00 0.00 0.0 0.00 V2 0.72 1.00 0.56 0.00 0.00 0.00 0.00 0.0 0.00 V3 0.63 0.56 1.00 0.00 0.00 0.00 0.00 0.0 0.00 V4 0.00 0.00 0.00 1.00 0.42 0.35 0.00 0.0 0.00 V5 0.00 0.00 0.00 0.42 1.00 0.30 0.00 0.0 0.00 V6 0.00 0.00 0.00 0.35 0.30 1.00 0.00 0.0 0.00 V7 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.3 0.24 V8 0.00 0.00 0.00 0.00 0.00 0.00 0.30 1.0 0.20 V9 0.00 0.00 0.00 0.00 0.00 0.00 0.24 0.2 1.00 $reliability (population reliability) [1] 0.81 0.64 0.49 0.49 0.36 0.25 0.36 0.25 0.16 2.4.3 f x is a matrix and Phi I is a correlated factors model > Phi = matrix(c(1,.5,.3,.5,1,.2,.3,.2,1), ncol=3) > cor.f3 <- sim.structure(fx,phi) > fx [,1] [,2] [,3] [1,] 0.9 0.0 0.0 [2,] 0.8 0.0 0.0 [3,] 0.7 0.0 0.0 [4,] 0.0 0.7 0.0 [5,] 0.0 0.6 0.0 [6,] 0.0 0.5 0.0 [7,] 0.0 0.0 0.6 [8,] 0.0 0.0 0.5 [9,] 0.0 0.0 0.4 11

Structural model x1 x2 x3 x4 x5 x6 x7 x8 x9 0.9 0.8 0.7 0.7 0.6 0.5 0.6 0.5 0.4 X1 X2 X3 Figure 4: Three uncorrelated factors generated using the sim.structure function and drawn using structure.diagram. 12

> Phi [,1] [,2] [,3] [1,] 1.0 0.5 0.3 [2,] 0.5 1.0 0.2 [3,] 0.3 0.2 1.0 > cor.f3 Call: sim.structure(fx = fx, Phi = Phi) $model (Population correlation matrix) V1 V2 V3 V4 V5 V6 V7 V8 V9 V1 1.00 0.720 0.630 0.315 0.270 0.23 0.162 0.14 0.108 V2 0.72 1.000 0.560 0.280 0.240 0.20 0.144 0.12 0.096 V3 0.63 0.560 1.000 0.245 0.210 0.17 0.126 0.10 0.084 V4 0.32 0.280 0.245 1.000 0.420 0.35 0.084 0.07 0.056 V5 0.27 0.240 0.210 0.420 1.000 0.30 0.072 0.06 0.048 V6 0.23 0.200 0.175 0.350 0.300 1.00 0.060 0.05 0.040 V7 0.16 0.144 0.126 0.084 0.072 0.06 1.000 0.30 0.240 V8 0.14 0.120 0.105 0.070 0.060 0.05 0.300 1.00 0.200 V9 0.11 0.096 0.084 0.056 0.048 0.04 0.240 0.20 1.000 $reliability (population reliability) [1] 0.81 0.64 0.49 0.49 0.36 0.25 0.36 0.25 0.16 Using symbolic loadings and path coefficients For some purposes, it is helpful not to specify particular values for the paths, but rather to think of them symbolically. This can be shown with symbolic loadings and path coefficients by using the structure.list and phi.list functions to create the fx and Phi matrices (Figure 5). > fxs <- structure.list(9,list(f1=c(1,2,3),f2=c(4,5,6),f3=c(7,8,9))) > Phis <- phi.list(3,list(f1=c(2,3),f2=c(1,3),f3=c(1,2))) > fxs #show the matrix F1 F2 F3 [1,] "a1" "0" "0" [2,] "a2" "0" "0" [3,] "a3" "0" "0" [4,] "0" "b4" "0" [5,] "0" "b5" "0" [6,] "0" "b6" "0" [7,] "0" "0" "c7" [8,] "0" "0" "c8" [9,] "0" "0" "c9" > Phis #show this one as well F1 F2 F3 F1 "1" "rba" "rca" 13

F2 "rab" "1" "rcb" F3 "rac" "rbc" "1" The structure.list and phi.list functions allow for creation of fx, Phi, and fy matrices in a very compact form, just by specifying the relevant variables. > #plot.new() > corf3.mod <- structure.diagram(fxs,phis) Structural model x1 x2 x3 a1 a2 a3 rab F1 x4 x5 x6 b4 b5 b6 rbc F2 rac x7 x8 c7 c8 c9 F3 x9 Figure 5: Three correlated factors with symbolic paths. Created using structure.diagram and structure.list and phi.list for ease of input. Drawing path models from Exploratory Factor Analysis solutions Alternatively, this result can represent the estimated factor loadings and oblique correlations found using factanal (Maximum Likelihood factoring) or fa (Principal axis or minimum residual (minres) factoring) followed by a promax rotation using the Promax function (Figure 6. 14

Comparing this figure with the previous one (Figure 5), it will be seen that one path was dropped because it was less than the arbitrary cut value of.2. > f3.p <- Promax(fa(cor.f3$model,3)) > #plot.new() > mod.f3p <- structure.diagram(f3.p,cut=.2) Structural model V1 V2 V3 V4 V5 V6 V7 V8 V9 1 0.7 0.6 0.4 0.5 0.9 MR3 MR1 MR2 Figure 6: The empirically fitted structural model. Paths less than cut (.2 in this case, the default is.3) are not shown. 2.4.4 f x and f y are matrices, and Phi I represents their correlations A more complicated model is when there is a f y vector or matrix representing a set of Y latent variables that are associated with the a set of y variables. In this case, the Phi matrix is a set of correlations within the X set and between the X and Y set. 15

> set.seed(42) > fx <- matrix(c(.9,.8,.7,rep(0,9),.7,.6,.5,rep(0,9),.6,.5,.4), ncol=3) > fy <- c(.6,.5,.4) > Phi <- matrix(c(1,.48,.32,.4,.48,1,.32,.3,.32,.32,1,.2,.4,.3,.2,1), ncol=4) > twelvev <- sim.structure(fx,phi, fy)$model > colnames(twelvev) <-rownames(twelvev) <- c(paste("x",1:9,sep=""),paste("y",1:3,sep="")) > round(twelvev,2) x1 x2 x3 x4 x5 x6 x7 x8 x9 y1 y2 y3 x1 1.00 0.72 0.63 0.30 0.26 0.22 0.17 0.14 0.12 0.22 0.18 0.14 x2 0.72 1.00 0.56 0.27 0.23 0.19 0.15 0.13 0.10 0.19 0.16 0.13 x3 0.63 0.56 1.00 0.24 0.20 0.17 0.13 0.11 0.09 0.17 0.14 0.11 x4 0.30 0.27 0.24 1.00 0.42 0.35 0.13 0.11 0.09 0.13 0.10 0.08 x5 0.26 0.23 0.20 0.42 1.00 0.30 0.12 0.10 0.08 0.11 0.09 0.07 x6 0.22 0.19 0.17 0.35 0.30 1.00 0.10 0.08 0.06 0.09 0.08 0.06 x7 0.17 0.15 0.13 0.13 0.12 0.10 1.00 0.30 0.24 0.07 0.06 0.05 x8 0.14 0.13 0.11 0.11 0.10 0.08 0.30 1.00 0.20 0.06 0.05 0.04 x9 0.12 0.10 0.09 0.09 0.08 0.06 0.24 0.20 1.00 0.05 0.04 0.03 y1 0.22 0.19 0.17 0.13 0.11 0.09 0.07 0.06 0.05 1.00 0.30 0.24 y2 0.18 0.16 0.14 0.10 0.09 0.08 0.06 0.05 0.04 0.30 1.00 0.20 y3 0.14 0.13 0.11 0.08 0.07 0.06 0.05 0.04 0.03 0.24 0.20 1.00 Data with this structure may be created using the sim.structure function, and shown either with the numeric values or symbolically using the structure.diagram function (Figure 7). > fxs <- structure.list(9,list(x1=c(1,2,3), X2 =c(4,5,6),x3 = c(7,8,9))) > phi <- phi.list(4,list(f1=c(4),f2=c(4),f3=c(4),f4=c(1,2,3))) > fyx <- structure.list(3,list(y=c(1,2,3)),"y") 2.4.5 A hierarchical structure among the latent predictors. Measures of intelligence and psychopathology frequently have a general factor as well as multiple group factors. The general factor then is thought to predict some dependent latent variable. Compare this with the previous model (see Figure 7). These two models can be compared using structural modeling procedures (see below). 3 Exploratory functions for analyzing structure Given correlation matrices such as those seen above for congeneric or bifactor models, the question becomes how best to estimate the underlying structure. Because these data sets were generated from a known model, the question becomes how well does a particular model recover the underlying structure. 16

> #plot.new() > sg3 <- structure.diagram(fxs,phi,fyx) Structural model x1 x2 x3 a1 a2 a3 X1 x4 x5 x6 b4 b5 b6 X2 rad rbd rcd Y Ya1 Ya2 Ya3 y1 y2 y3 x7 x8 c7 c8 c9 X3 x9 Figure 7: A symbolic structural model. Three independent latent variables are regressed on a latent Y. 17

> fxh <- structure.list(9,list(x1=c(1:3),x2=c(4:6),x3=c(7:9),g=null)) > fy <- structure.list(3,list(y=c(1,2,3))) > Phi <- diag(1,5,5) > Phi[4,c(1:3)] <- letters[1:3] > Phi[5,4] <- "r" > #plot.new() > hi.mod <-structure.diagram(fxh,phi, fy) Structural model x1 x2 x3 a1 a2 a3 X1 x4 x5 x6 x7 x8 b4 b5 b6 c7 c8 c9 X2 X3 c g b a r Y a1 a2 a3 y1 y2 y3 x9 Figure 8: A symbolic structural model with a general factor and three group factors. The general factor is regressed on the latent Y variable. 18

3.1 Exploratory simple structure models The technique of principal components provides a set of weighted linear composites that best aproximates a particular correlation or covariance matrix. If these are then rotated to provide a more interpretable solution, the components are no longer the principal components. The principal function will extract the first n principal components (default value is 1) and if n>1, rotate to simple structure using a varimax, quartimin, or Promax criterion. > principal(cong1$model) Principal Components Analysis Call: principal(r = cong1$model) Standardized loadings (pattern matrix) based upon correlation matrix PC1 h2 u2 com V1 0.89 0.80 0.20 1 V2 0.85 0.73 0.27 1 V3 0.80 0.64 0.36 1 V4 0.73 0.53 0.47 1 PC1 SS loadings 2.69 Proportion Var 0.67 Mean item complexity = 1 Test of the hypothesis that 1 component is sufficient. The root mean square of the residuals (RMSR) is 0.11 Fit based upon off diagonal values = 0.96 > fa(cong1$model) Factor Analysis using method = minres Call: fa(r = cong1$model) Standardized loadings (pattern matrix) based upon correlation matrix MR1 h2 u2 com V1 0.9 0.81 0.19 1 V2 0.8 0.64 0.36 1 V3 0.7 0.49 0.51 1 V4 0.6 0.36 0.64 1 MR1 SS loadings 2.30 Proportion Var 0.57 Mean item complexity = 1 Test of the hypothesis that 1 factor is sufficient. The degrees of freedom for the null model are 6 and the objective function was 1.65 The degrees of freedom for the model are 2 and the objective function was 0 The root mean square of the residuals (RMSR) is 0 The df corrected root mean square of the residuals is 0 Fit based upon off diagonal values = 1 19

Measures of factor score adequacy MR1 Correlation of scores with factors 0.94 Multiple R square of scores with factors 0.88 Minimum correlation of possible factor scores 0.77 It is important to note that although the principal components function does not exactly reproduce the model parameters, the factor.pa function, implementing principal axes or minimum residual (minres) factor analysis, does. Consider the case of three underlying factors as seen in the bifact example above. Because the number of observations is not specified, there is no associated χ 2 value. The factor.congruence function reports the cosine of the angle between the factors. > pc3 <- principal(bifact,3) > pa3 <- fa(bifact,3,fm="pa") > ml3 <- fa(bifact,3,fm="ml") > pc3 Principal Components Analysis Call: principal(r = bifact, nfactors = 3) Standardized loadings (pattern matrix) based upon correlation matrix PC1 PC3 PC2 h2 u2 com V1 0.75 0.27 0.21 0.69 0.31 1.4 V2 0.76 0.21 0.16 0.64 0.36 1.2 V3 0.78 0.11 0.10 0.63 0.37 1.1 V4 0.29 0.69 0.15 0.59 0.41 1.5 V5 0.20 0.71 0.11 0.56 0.44 1.2 V6 0.07 0.76 0.08 0.59 0.41 1.0 V7 0.26 0.16 0.70 0.58 0.42 1.4 V8 0.20 0.11 0.71 0.55 0.45 1.2 V9 0.00 0.06 0.73 0.53 0.47 1.0 PC1 PC3 PC2 SS loadings 1.99 1.73 1.64 Proportion Var 0.22 0.19 0.18 Cumulative Var 0.22 0.41 0.60 Proportion Explained 0.37 0.32 0.31 Cumulative Proportion 0.37 0.69 1.00 Mean item complexity = 1.2 Test of the hypothesis that 3 components are sufficient. The root mean square of the residuals (RMSR) is 0.1 Fit based upon off diagonal values = 0.88 > pa3 Factor Analysis using method = pa Call: fa(r = bifact, nfactors = 3, fm = "pa") Standardized loadings (pattern matrix) based upon correlation matrix PA1 PA3 PA2 h2 u2 com V1 0.8 0.0 0.00 0.64 0.36 1 V2 0.7 0.0 0.00 0.49 0.51 1 V3 0.6 0.0 0.00 0.36 0.64 1 V4 0.0 0.7 0.00 0.49 0.51 1 20

V5 0.0 0.6 0.00 0.36 0.64 1 V6 0.0 0.5 0.00 0.25 0.75 1 V7 0.0 0.0 0.69 0.48 0.52 1 V8 0.0 0.0 0.61 0.36 0.64 1 V9 0.0 0.0 0.40 0.16 0.84 1 PA1 PA3 PA2 SS loadings 1.49 1.10 1.01 Proportion Var 0.17 0.12 0.11 Cumulative Var 0.17 0.29 0.40 Proportion Explained 0.41 0.31 0.28 Cumulative Proportion 0.41 0.72 1.00 With factor correlations of PA1 PA3 PA2 PA1 1.00 0.72 0.63 PA3 0.72 1.00 0.56 PA2 0.63 0.56 1.00 Mean item complexity = 1 Test of the hypothesis that 3 factors are sufficient. The degrees of freedom for the null model are 36 and the objective function was 1.88 The degrees of freedom for the model are 12 and the objective function was 0 The root mean square of the residuals (RMSR) is 0 The df corrected root mean square of the residuals is 0 Fit based upon off diagonal values = 1 Measures of factor score adequacy PA1 PA3 PA2 Correlation of scores with factors 0.9 0.85 0.83 Multiple R square of scores with factors 0.8 0.72 0.69 Minimum correlation of possible factor scores 0.6 0.45 0.38 > ml3 Factor Analysis using method = ml Call: fa(r = bifact, nfactors = 3, fm = "ml") Standardized loadings (pattern matrix) based upon correlation matrix ML1 ML3 ML2 h2 u2 com V1 0.8 0.0 0.0 0.64 0.36 1 V2 0.7 0.0 0.0 0.49 0.51 1 V3 0.6 0.0 0.0 0.36 0.64 1 V4 0.0 0.7 0.0 0.49 0.51 1 V5 0.0 0.6 0.0 0.36 0.64 1 V6 0.0 0.5 0.0 0.25 0.75 1 V7 0.0 0.0 0.7 0.49 0.51 1 V8 0.0 0.0 0.6 0.36 0.64 1 V9 0.0 0.0 0.4 0.16 0.84 1 ML1 ML3 ML2 SS loadings 1.49 1.10 1.01 Proportion Var 0.17 0.12 0.11 Cumulative Var 0.17 0.29 0.40 Proportion Explained 0.41 0.31 0.28 Cumulative Proportion 0.41 0.72 1.00 21

With factor correlations of ML1 ML3 ML2 ML1 1.00 0.72 0.63 ML3 0.72 1.00 0.56 ML2 0.63 0.56 1.00 Mean item complexity = 1 Test of the hypothesis that 3 factors are sufficient. The degrees of freedom for the null model are 36 and the objective function was 1.88 The degrees of freedom for the model are 12 and the objective function was 0 The root mean square of the residuals (RMSR) is 0 The df corrected root mean square of the residuals is 0 Fit based upon off diagonal values = 1 Measures of factor score adequacy ML1 ML3 ML2 Correlation of scores with factors 0.90 0.85 0.83 Multiple R square of scores with factors 0.80 0.72 0.69 Minimum correlation of possible factor scores 0.61 0.45 0.38 > factor.congruence(list(pc3,pa3,ml3)) PC1 PC3 PC2 PA1 PA3 PA2 ML1 ML3 ML2 PC1 1.00 0.49 0.42 0.93 0.24 0.21 0.93 0.24 0.21 PC3 0.49 1.00 0.35 0.27 0.94 0.15 0.27 0.93 0.15 PC2 0.42 0.35 1.00 0.22 0.16 0.94 0.22 0.16 0.94 PA1 0.93 0.27 0.22 1.00 0.00 0.00 1.00 0.00 0.00 PA3 0.24 0.94 0.16 0.00 1.00 0.00 0.00 1.00 0.00 PA2 0.21 0.15 0.94 0.00 0.00 1.00 0.00 0.00 1.00 ML1 0.93 0.27 0.22 1.00 0.00 0.00 1.00 0.00 0.00 ML3 0.24 0.93 0.16 0.00 1.00 0.00 0.00 1.00 0.00 ML2 0.21 0.15 0.94 0.00 0.00 1.00 0.00 0.00 1.00 By default, all three of these procedures use the varimax rotation criterion. Perhaps it is useful to apply an oblique transformation such as Promax or oblimin to the results. The Promax function in psych differs slightly from the standard promax in that it reports the factor intercorrelations. > ml3p <- Promax(ml3) > ml3p Call: NULL Standardized loadings (pattern matrix) based upon correlation matrix ML1 ML3 ML2 h2 u2 V1 0.8 0.0 0.0 0.64 0.36 V2 0.7 0.0 0.0 0.49 0.51 V3 0.6 0.0 0.0 0.36 0.64 V4 0.0 0.7 0.0 0.49 0.51 V5 0.0 0.6 0.0 0.36 0.64 V6 0.0 0.5 0.0 0.25 0.75 V7 0.0 0.0 0.7 0.49 0.51 V8 0.0 0.0 0.6 0.36 0.64 V9 0.0 0.0 0.4 0.16 0.84 ML1 ML3 ML2 SS loadings 1.49 1.10 1.01 Proportion Var 0.17 0.12 0.11 22

Cumulative Var 0.17 0.29 0.40 Proportion Explained 0.41 0.31 0.28 Cumulative Proportion 0.41 0.72 1.00 ML1 ML3 ML2 ML1 1 0 0 ML3 0 1 0 ML2 0 0 1 3.2 Exploratory hierarchical models In addition to the conventional oblique factor model, an alternative model is to consider the correlations between the factors to represent a higher order factor. This can be shown either as a bifactor solution Holzinger and Swineford (1937); Schmid and Leiman (1957) with a general factor for all variables and a set of residualized group factors, or as a hierarchical structure. An exploratory hierarchical model can be applied to this kind of data structure using the omega function. Graphic options include drawing a Schmid - Leiman bifactor solution (Figure 9) or drawing a hierarchical factor solution f(figure 10). 3.2.1 A bifactor solution The bifactor solution has a general factor loading for each variable as well as a set of residual group factors. This approach has been used extensively in the measurement of ability and has more recently been used in the measure of psychopathology (Reise et al., 2007). Data sets included in the bifactor data include the original (Holzinger and Swineford, 1937) data set (holzinger) as well as a set from Reise et al. (2007) (reise) and a nine variable problem from Thurstone. 3.2.2 A hierarchical solution Both of these graphical representations are reflected in the output of the omega function. The first was done using a Schmid-Leiman transformation, the second was not. As will be seen later, the objects returned from these two analyses may be used as models for a sem analysis. It is also useful to examine the estimates of reliability reported by omega. > om.bi Omega Call: omega(m = bifact) Alpha: 0.78 G.6: 0.78 Omega Hierarchical: 0.7 Omega H asymptotic: 0.85 23

> om.bi <- omega(bifact) Omega V1 0.7 0.6 V2 V3 0.3 0.3 0.3 F1* 0.5 0.6 V4 0.4 g 0.5 V5 0.4 F2* 0.4 V6 0.3 0.5 0.4 0.3 V7 V8 0.5 0.4 0.3 F3* V9 Figure 9: An exploratory bifactor solution to the nine variable problem 24

> om.hi <- omega(bifact,sl=false) Omega V1 V2 V3 0.8 0.7 0.6 F1 V4 0.7 0.9 V5 0.6 F2 0.8 g V6 0.5 0.7 V7 V8 V9 0.7 0.6 0.4 F3 Figure 10: An exploratory hierarchical solution to the nine variable problem. 25

Omega Total 0.82 Schmid Leiman Factor loadings greater than 0.2 g F1* F2* F3* h2 u2 p2 V1 0.72 0.35 0.64 0.36 0.81 V2 0.63 0.31 0.49 0.51 0.81 V3 0.54 0.26 0.36 0.64 0.81 V4 0.56 0.42 0.49 0.51 0.64 V5 0.48 0.36 0.36 0.64 0.64 V6 0.40 0.30 0.25 0.75 0.64 V7 0.49 0.50 0.49 0.51 0.49 V8 0.42 0.43 0.36 0.64 0.49 V9 0.28 0.29 0.16 0.84 0.49 With eigenvalues of: g F1* F2* F3* 2.41 0.28 0.40 0.52 general/max 4.67 max/min = 1.82 mean percent general = 0.65 with sd = 0.14 and cv of 0.21 Explained Common Variance of the general factor = 0.67 The degrees of freedom are 12 and the fit is 0 The root mean square of the residuals is 0 The df corrected root mean square of the residuals is 0 Compare this with the adequacy of just a general factor and no group factors The degrees of freedom for just the general factor are 27 and the fit is 0.23 The root mean square of the residuals is 0.07 The df corrected root mean square of the residuals is 0.08 Measures of factor score adequacy g F1* F2* F3* Correlation of scores with factors 0.86 0.47 0.57 0.64 Multiple R square of scores with factors 0.74 0.22 0.33 0.41 Minimum correlation of factor score estimates 0.47-0.56-0.35-0.18 Total, General and Subset omega for each subset g F1* F2* F3* 26

Omega total for total scores and subscales 0.82 0.74 0.63 0.59 Omega general for total scores and subscales 0.70 0.60 0.40 0.29 Omega group for total scores and subscales 0.12 0.14 0.23 0.30 Yet one more way to treat the hierarchical structure of a data set is to consider hierarchical cluster analysis using the ICLUST algorithm (Figure 11). ICLUST is most appropriate for forming item composites. Hierarchical cluster analysis of bifact data V9 V8 V7 V6 V5 V4 V3 V2 V1 0.75 0.75 0.65 0.65 0.65 0.65 0.78 0.64 C2 α = 0.59 β = 0.59 0.69 C1 α = 0.72 β = 0.72 C3 α = 0.59 β = 0.59 0.86 0.84 0.72 C4 α = 0.74 β = 0.68 C5 α = 0.62 β = 0.57 0.73 0.75 C6 α = 0.76 β = 0.66 C7 α = 0.58 β = 0.48 0.7 0.65 C8 α = 0.78 β = 0.61 Figure 11: A hierarchical cluster analysis of the bifact data set using ICLUST 27

4 Confirmatory models Although the exploratory models shown above do estimate the goodness of fit of the model and compare the residual matrix to a zero matrix using a χ 2 statistic, they estimate more parameters than are necessary if there is indeed a simple structure, and they do not allow for tests of competing models. The sem function in the sem package by John Fox allows for confirmatory tests. The interested reader is referred to the sem manual for more detail (Fox et al., 2013). 4.1 Using psych as a front end for the sem package Because preparation of the sem commands is a bit tedious, several of the psych package functions have been designed to provide the appropriate commands. That is, the functions structure.list, phi.list, structure.diagram, structure.sem, and omega.graph may be used as a front end to sem. Usually with no modification, but sometimes with just slight modification, the model output from the structure.diagram, structure.sem, and omega.graph functions is meant to provide the appropriate commands for sem. 4.2 Testing a congeneric model versus a tau equivalent model The congeneric model is a one factor model with possibly unequal factor loadings. The tau equivalent model model is one with equal factor loadings. Tests for these may be done by creating the appropriate structures. The structure.graph function which requires Rgraphviz, or structure.diagram or the structure.sem functions which do not may be used. The following example tests the hypothesis (which is actually false) that the correlations found in the cong data set (see 2.1) are tau equivalent. Because the variable labels in that data set were V1... V4, we specify the labels to match those. > library(sem) > mod.tau <- structure.sem(c("a","a","a","a"),labels=paste("v",1:4,sep="")) > mod.tau #show it Path Parameter Value [1,] "X1->V1" "a" NA [2,] "X1->V2" "a" NA [3,] "X1->V3" "a" NA [4,] "X1->V4" "a" NA [5,] "V1<->V1" "x1e" NA [6,] "V2<->V2" "x2e" NA [7,] "V3<->V3" "x3e" NA [8,] "V4<->V4" "x4e" NA [9,] "X1<->X1" NA "1" attr(,"class") [1] "mod" 28

> sem.tau <- sem(mod.tau,cong,100) > summary(sem.tau,digits=2) Model Chisquare = 6.593496 Df = 5 Pr(>Chisq) = 0.2526696 AIC = 16.5935 BIC = -16.43236 Normalized Residuals Min. 1st Qu. Median Mean 3rd Qu. Max. -1.03200-0.44200-0.25030-0.07905 0.52700 0.88770 R-square for Endogenous Variables V1 V2 V3 V4 0.5245 0.4592 0.4500 0.4432 Parameter Estimates Estimate Std Error z value Pr(> z ) a 0.6865481 0.06299180 10.899007 1.165221e-27 V1 <--- X1 x1e 0.4272839 0.08086561 5.283876 1.264786e-07 V1 <--> V1 x2e 0.5551772 0.09751222 5.693411 1.245260e-08 V2 <--> V2 x3e 0.5760999 0.10030974 5.743210 9.289853e-09 V3 <--> V3 x4e 0.5920607 0.10245375 5.778809 7.523134e-09 V4 <--> V4 Iterations = 11 Test whether the data are congeneric. That is, whether a one factor model fits. Compare this to the prior model using the anova function. > mod.cong <- structure.sem(c("a","b","c","d"),labels=paste("v",1:4,sep="")) > mod.cong #show the model Path Parameter Value [1,] "X1->V1" "a" NA [2,] "X1->V2" "b" NA [3,] "X1->V3" "c" NA [4,] "X1->V4" "d" NA [5,] "V1<->V1" "x1e" NA [6,] "V2<->V2" "x2e" NA [7,] "V3<->V3" "x3e" NA [8,] "V4<->V4" "x4e" NA [9,] "X1<->X1" NA "1" attr(,"class") [1] "mod" > sem.cong <- sem(mod.cong,cong,100) > summary(sem.cong,digits=2) Model Chisquare = 2.941678 Df = 2 Pr(>Chisq) = 0.2297327 AIC = 18.94168 BIC = -6.268663 Normalized Residuals Min. 1st Qu. Median Mean 3rd Qu. Max. -0.5739-0.0699 0.0339 0.0113 0.1605 0.5412 R-square for Endogenous Variables V1 V2 V3 V4 0.6880 0.4384 0.3942 0.3524 29

Parameter Estimates Estimate Std Error z value Pr(> z ) a 0.8294562 0.09786772 8.475279 2.345174e-17 V1 <--- X1 b 0.6621164 0.10066777 6.577243 4.792500e-11 V2 <--- X1 c 0.6278767 0.10146860 6.187891 6.097433e-10 V3 <--- X1 d 0.5936695 0.10238816 5.798224 6.702094e-09 V4 <--- X1 x1e 0.3120026 0.10044870 3.106089 1.895798e-03 V1 <--> V1 x2e 0.5616018 0.10154893 5.530356 3.195810e-08 V2 <--> V2 x3e 0.6057707 0.10421285 5.812822 6.142832e-09 V3 <--> V3 x4e 0.6475566 0.10732995 6.033326 1.606191e-09 V4 <--> V4 Iterations = 12 > anova(sem.cong,sem.tau) #test the difference between the two models LR Test for Difference Between Models Model Df Model Chisq Df LR Chisq Pr(>Chisq) sem.cong 2 2.9417 sem.tau 5 6.5935 3 3.6518 0.3016 The anova comparison of the congeneric versus tau equivalent model shows that the change in χ 2 is significant given the change in degrees of freedom. 4.3 Testing the dimensionality of a hierarchical data set by creating the model The bifact correlation matrix was created to represent a hierarchical structure. Various confirmatory models can be applied to this matrix. The first example creates the model directly, the next several create models based upon exploratory factor analyses. mod.one is a congeneric model of one factor accounting for the relationships between the nine variables. Although not correct, with 100 subjects, this model can not be rejected. However, an examination of the residuals suggests serious problems with the model. > mod.one <- structure.sem(letters[1:9],labels=paste("v",1:9,sep="")) > mod.one #show the model Path Parameter Value [1,] "X1->V1" "a" NA [2,] "X1->V2" "b" NA [3,] "X1->V3" "c" NA [4,] "X1->V4" "d" NA [5,] "X1->V5" "e" NA [6,] "X1->V6" "f" NA [7,] "X1->V7" "g" NA [8,] "X1->V8" "h" NA [9,] "X1->V9" "i" NA [10,] "V1<->V1" "x1e" NA [11,] "V2<->V2" "x2e" NA [12,] "V3<->V3" "x3e" NA [13,] "V4<->V4" "x4e" NA 30

[14,] "V5<->V5" "x5e" NA [15,] "V6<->V6" "x6e" NA [16,] "V7<->V7" "x7e" NA [17,] "V8<->V8" "x8e" NA [18,] "V9<->V9" "x9e" NA [19,] "X1<->X1" NA "1" attr(,"class") [1] "mod" > sem.one <- sem(mod.one,bifact,100) > summary(sem.one,digits=2) Model Chisquare = 21.16848 Df = 27 Pr(>Chisq) = 0.778334 AIC = 57.16848 BIC = -103.1711 Normalized Residuals Min. 1st Qu. Median Mean 3rd Qu. Max. -0.3337000-0.2924000-0.1940000 0.0369500 0.0000019 1.8880000 R-square for Endogenous Variables V1 V2 V3 V4 V5 V6 V7 V8 V9 0.5636 0.4524 0.3377 0.3292 0.2522 0.1798 0.2568 0.1980 0.0932 Parameter Estimates Estimate Std Error z value Pr(> z ) a 0.7507129 0.09512871 7.891549 2.984584e-15 V1 <--- X1 b 0.6726412 0.09807150 6.858682 6.949882e-12 V2 <--- X1 c 0.5811209 0.10137932 5.732145 9.916850e-09 V3 <--- X1 d 0.5737425 0.10163173 5.645309 1.648847e-08 V4 <--- X1 e 0.5021915 0.10392785 4.832116 1.350893e-06 V5 <--- X1 f 0.4239908 0.10609489 3.996335 6.433059e-05 V6 <--- X1 g 0.5067957 0.10378883 4.882950 1.045102e-06 V7 <--- X1 h 0.4450171 0.10554867 4.216227 2.484236e-05 V8 <--- X1 i 0.3052415 0.10867168 2.808841 4.972015e-03 V9 <--- X1 x1e 0.4364302 0.08884720 4.912143 9.008624e-07 V1 <--> V1 x2e 0.5475539 0.09653118 5.672300 1.408927e-08 V2 <--> V2 x3e 0.6622982 0.10678972 6.201891 5.578883e-10 V3 <--> V3 x4e 0.6708197 0.10761127 6.233731 4.554546e-10 V4 <--> V4 x5e 0.7478036 0.11527456 6.487153 8.747374e-11 V5 <--> V5 x6e 0.8202314 0.12278021 6.680485 2.381530e-11 V6 <--> V6 x7e 0.7431581 0.11480162 6.473411 9.581477e-11 V7 <--> V7 x8e 0.8019593 0.12086557 6.635134 3.242077e-11 V8 <--> V8 x9e 0.9068284 0.13200380 6.869714 6.433079e-12 V9 <--> V9 Iterations = 11 > round(residuals(sem.one),2) V1 V2 V3 V4 V5 V6 V7 V8 V9 V1 0.00 0.06 0.04-0.03-0.03-0.03-0.03-0.03-0.03 V2 0.06 0.00 0.03-0.03-0.04-0.03-0.03-0.03-0.03 V3 0.04 0.03 0.00-0.03-0.03-0.03-0.03-0.03-0.03 V4-0.03-0.03-0.03 0.00 0.13 0.11-0.02-0.02-0.02 V5-0.03-0.04-0.03 0.13 0.00 0.09-0.02-0.02-0.02 V6-0.03-0.03-0.03 0.11 0.09 0.00-0.02-0.02-0.02 V7-0.03-0.03-0.03-0.02-0.02-0.02 0.00 0.19 0.13 V8-0.03-0.03-0.03-0.02-0.02-0.02 0.19 0.00 0.10 V9-0.03-0.03-0.03-0.02-0.02-0.02 0.13 0.10 0.00 31

4.4 Testing the dimensionality based upon an exploratory analysis Alternatively, the output from an exploratory factor analysis can be used as input to the structure.sem function. > f1 <- factanal(covmat=bifact,factors=1) > mod.f1 <- structure.sem(f1) > sem.f1 <- sem(mod.f1,bifact,100) > sem.f1 Model Chisquare = 21.16848 Df = 27 V1 V2 V3 V4 V5 V6 V7 V8 0.7507129 0.6726412 0.5811209 0.5737425 0.5021915 0.4239908 0.5067957 0.4450171 V9 x1e x2e x3e x4e x5e x6e x7e 0.3052415 0.4364302 0.5475539 0.6622982 0.6708197 0.7478036 0.8202314 0.7431581 x8e x9e 0.8019593 0.9068284 Iterations = 11 The answers are, of course, identical. 4.5 Specifying a three factor model An alternative model is to extract three factors and try this solution. The fa factor analysis function (using the minimum residual algorithm) is used to detect the structure. Alternatively, the factanal could have been used. Rather than use the default rotation of oblimin, we force an orthogonal solution (even though we know it will be a poor solution). > f3 <-fa(bifact,3,rotate="varimax") > mod.f3 <- structure.sem(f3) > sem.f3 <- sem(mod.f3,bifact,100) > summary(sem.f3,digits=2) Model Chisquare = 53.86635 Df = 27 Pr(>Chisq) = 0.001579738 AIC = 89.86635 BIC = -70.47325 Normalized Residuals Min. 1st Qu. Median Mean 3rd Qu. Max. -0.000003 0.000000 1.950000 1.642000 2.633000 4.012000 R-square for Endogenous Variables V1 V2 V3 V4 V5 V6 V7 V8 V9 0.64 0.49 0.36 0.49 0.36 0.25 0.49 0.36 0.16 Parameter Estimates Estimate Std Error z value Pr(> z ) F1V1 0.8000000 0.1114517 7.177994 7.074151e-13 V1 <--- MR1 F1V2 0.7000001 0.1089845 6.422931 1.336754e-10 V2 <--- MR1 F1V3 0.6000000 0.1068002 5.617968 1.932167e-08 V3 <--- MR1 F2V4 0.6999999 0.1427544 4.903527 9.413091e-07 V4 <--- MR3 32

F2V5 0.6000001 0.1328610 4.515998 6.301927e-06 V5 <--- MR3 F2V6 0.4999995 0.1238740 4.036354 5.428827e-05 V6 <--- MR3 F3V7 0.7000001 0.1680059 4.166522 3.092827e-05 V7 <--- MR2 F3V8 0.6000000 0.1530271 3.920873 8.822871e-05 V8 <--- MR2 F3V9 0.4000005 0.1265677 3.160368 1.575701e-03 V9 <--- MR2 x1e 0.3600000 0.1297434 2.774707 5.525146e-03 V1 <--> V1 x2e 0.5099999 0.1165643 4.375268 1.212834e-05 V2 <--> V2 x3e 0.6399999 0.1130156 5.662936 1.488043e-08 V3 <--> V3 x4e 0.5100000 0.1739239 2.932316 3.364440e-03 V4 <--> V4 x5e 0.6399998 0.1475345 4.337967 1.438068e-05 V5 <--> V5 x6e 0.7500007 0.1336788 5.610466 2.017821e-08 V6 <--> V6 x7e 0.5100000 0.2136118 2.387509 1.696298e-02 V7 <--> V7 x8e 0.6400001 0.1734024 3.690837 2.235172e-04 V8 <--> V8 x9e 0.8400000 0.1362332 6.165898 7.008420e-10 V9 <--> V9 Iterations = 24 > round(residuals(sem.f3),2) V1 V2 V3 V4 V5 V6 V7 V8 V9 V1 0.00 0.00 0.00 0.40 0.35 0.29 0.35 0.30 0.20 V2 0.00 0.00 0.00 0.35 0.30 0.25 0.31 0.26 0.18 V3 0.00 0.00 0.00 0.30 0.26 0.22 0.26 0.23 0.15 V4 0.40 0.35 0.30 0.00 0.00 0.00 0.27 0.24 0.16 V5 0.35 0.30 0.26 0.00 0.00 0.00 0.24 0.20 0.13 V6 0.29 0.25 0.22 0.00 0.00 0.00 0.20 0.17 0.11 V7 0.35 0.31 0.26 0.27 0.24 0.20 0.00 0.00 0.00 V8 0.30 0.26 0.23 0.24 0.20 0.17 0.00 0.00 0.00 V9 0.20 0.18 0.15 0.16 0.13 0.11 0.00 0.00 0.00 The residuals show serious problems with this model. Although the residuals within each of the three factors are zero, the residuals between groups are much too large. 4.6 Allowing for an oblique solution The previous solution is clearly very bad. What would happen if the exploratory solution were allowed to have correlated (oblique) factors? > f3 <-fa(bifact,3) #extract three factors and do an oblique rotation > mod.f3 <- structure.sem(f3) #create the sem model > mod.f3 #show it Path Parameter Value [1,] "MR1->V1" "F1V1" NA [2,] "MR1->V2" "F1V2" NA [3,] "MR1->V3" "F1V3" NA [4,] "MR3->V4" "F2V4" NA [5,] "MR3->V5" "F2V5" NA [6,] "MR3->V6" "F2V6" NA [7,] "MR2->V7" "F3V7" NA [8,] "MR2->V8" "F3V8" NA [9,] "MR2->V9" "F3V9" NA [10,] "V1<->V1" "x1e" NA [11,] "V2<->V2" "x2e" NA [12,] "V3<->V3" "x3e" NA 33

[13,] "V4<->V4" "x4e" NA [14,] "V5<->V5" "x5e" NA [15,] "V6<->V6" "x6e" NA [16,] "V7<->V7" "x7e" NA [17,] "V8<->V8" "x8e" NA [18,] "V9<->V9" "x9e" NA [19,] "MR3<->MR1" "rf2f1" NA [20,] "MR2<->MR1" "rf3f1" NA [21,] "MR2<->MR3" "rf3f2" NA [22,] "MR1<->MR1" NA "1" [23,] "MR3<->MR3" NA "1" [24,] "MR2<->MR2" NA "1" attr(,"class") [1] "mod" The structure being tested may be seen using structure.graph Structural model V1 V2 V3 0.8 0.7 0.6 MR1 V4 V5 V6 0.7 0.6 0.5 MR3 0.7 0.6 0.6 V7 V8 0.7 0.6 0.4 MR2 V9 Figure 12: A three factor, oblique solution. 34

This makes much better sense, and in fact (as hoped) recovers the original structure. 4.7 Extract a bifactor solution using omega and then test that model using sem A bifactor solution has previously been shown (Figure 9). The output from the omega function includes the sem commands for the analysis. As an example of doing this with real rather than simulated data, consider 9 variables from Thurstone. For completeness, the stdcoef from sem is used as well as the summary function. 4.7.1 sem of Thurstone 9 variable problem The sem manual includes an example of a hierarchical solution to 9 mental abilities originally reported by Thurstone and used in the SAS manual for PROC CALIS and discussed in detail by McDonald (1999). The data matrix, as reported by Fox may be found in the Thurstone data set (which is lazy loaded ). Using the commands just shown, it is possible to analyze this data set using a bifactor solution (Figure 13). > sem.bi <- sem(om.th.bi$model,thurstone,213) #use the model created by omega > summary(sem.bi,digits=2) Model Chisquare = 24.2163 Df = 18 Pr(>Chisq) = 0.1480685 AIC = 78.2163 BIC = -72.28696 Normalized Residuals Min. 1st Qu. Median Mean 3rd Qu. Max. -0.8212000-0.3341000-0.0000009 0.0281700 0.1562000 1.7970000 R-square for Endogenous Variables Sentences Vocabulary Sent.Completion First.Letters 4.Letter.Words 0.8276 0.8302 0.7315 0.7472 0.6126 Suffixes Letter.Series Pedigrees Letter.Group 0.4824 0.8503 0.4996 0.4483 Parameter Estimates Estimate Std Error z value Pr(> z ) Sentences 0.7678671 0.07059396 10.8772353 1.479833e-27 Vocabulary 0.7909248 0.06969232 11.3488087 7.518003e-30 Sent.Completion 0.7536211 0.07113218 10.5946585 3.154903e-26 First.Letters 0.6083814 0.07063841 8.6126138 7.141338e-18 4.Letter.Words 0.5973349 0.07092937 8.4215455 3.715499e-17 Suffixes 0.5717903 0.07157752 7.9884057 1.366950e-15 Letter.Series 0.5668949 0.07249339 7.8199523 5.284337e-15 Pedigrees 0.6623314 0.07003035 9.4577757 3.145633e-21 Letter.Group 0.5299524 0.07332494 7.2274501 4.921470e-13 F1*Sentences 0.4878698 0.08141095 5.9926801 2.064107e-09 F1*Vocabulary 0.4523234 0.08353995 5.4144562 6.147524e-08 F1*Sent.Completion 0.4044507 0.08727334 4.6342988 3.581494e-06 F2*First.Letters 0.6140531 0.08471145 7.2487623 4.205973e-13 35

Omega Sentences 0.71 0.73 0.68 0.65 Vocabulary 0.57 0.55 Sent.Completion 0.52 First.Letters 0.56 F1* g 0.62 4.Letter.Words 0.49 F2* 0.56 Suffixes 0.41 0.23 0.59 0.54 0.58 Letter.Series Letter.Group 0.61 0.46 0.34 F3* Pedigrees Figure 13: A bifactor solution to the Thurstone 9 variable problem. All items load on a general factor of ability, the residual factors account for the correlations between items within groups. 36

F2*4.Letter.Words 0.5058063 0.08145488 6.2096500 5.310276e-10 F2*Suffixes 0.3943208 0.07805383 5.0519075 4.374195e-07 F3*Letter.Series 0.7272955 0.15844866 4.5901015 4.430304e-06 F3*Pedigrees 0.2468417 0.08677536 2.8446053 4.446649e-03 F3*Letter.Group 0.4091495 0.11352380 3.6040854 3.132541e-04 e1 0.1723633 0.03405646 5.0611045 4.168346e-07 e2 0.1698419 0.03001233 5.6590697 1.521958e-08 e3 0.2684749 0.03316228 8.0957909 5.689350e-16 e4 0.2528108 0.07942835 3.1828791 1.458185e-03 e5 0.3873510 0.06317399 6.1314949 8.705712e-10 e6 0.5175679 0.05955079 8.6912013 3.586269e-18 e7 0.1496709 0.21861502 0.6846325 4.935759e-01 e8 0.5003855 0.05956551 8.4005902 4.442400e-17 e9 0.5517474 0.08455914 6.5249884 6.800680e-11 Sentences Sentences <--- g Vocabulary Vocabulary <--- g Sent.Completion Sent.Completion <--- g First.Letters First.Letters <--- g 4.Letter.Words 4.Letter.Words <--- g Suffixes Suffixes <--- g Letter.Series Letter.Series <--- g Pedigrees Pedigrees <--- g Letter.Group Letter.Group <--- g F1*Sentences Sentences <--- F1* F1*Vocabulary Vocabulary <--- F1* F1*Sent.Completion Sent.Completion <--- F1* F2*First.Letters First.Letters <--- F2* F2*4.Letter.Words 4.Letter.Words <--- F2* F2*Suffixes Suffixes <--- F2* F3*Letter.Series Letter.Series <--- F3* F3*Pedigrees Pedigrees <--- F3* F3*Letter.Group Letter.Group <--- F3* e1 Sentences <--> Sentences e2 Vocabulary <--> Vocabulary e3 Sent.Completion <--> Sent.Completion e4 First.Letters <--> First.Letters e5 4.Letter.Words <--> 4.Letter.Words e6 Suffixes <--> Suffixes e7 Letter.Series <--> Letter.Series e8 Pedigrees <--> Pedigrees e9 Letter.Group <--> Letter.Group Iterations = 72 > stdcoef(sem.bi,digits=2) Std. Estimate 1 Sentences 0.7678671 Sentences <--- g 2 Vocabulary 0.7909246 Vocabulary <--- g 3 Sent.Completion 0.7536211 Sent.Completion <--- g 4 First.Letters 0.6083814 First.Letters <--- g 5 4.Letter.Words 0.5973349 4.Letter.Words <--- g 6 Suffixes 0.5717900 Suffixes <--- g 7 Letter.Series 0.5668950 Letter.Series <--- g 8 Pedigrees 0.6623317 Pedigrees <--- g 9 Letter.Group 0.5299523 Letter.Group <--- g 10 F1*Sentences 0.4878697 Sentences <--- F1* 37

11 F1*Vocabulary 0.4523233 Vocabulary <--- F1* 12 F1*Sent.Completion 0.4044507 Sent.Completion <--- F1* 13 F2*First.Letters 0.6140531 First.Letters <--- F2* 14 F2*4.Letter.Words 0.5058063 4.Letter.Words <--- F2* 15 F2*Suffixes 0.3943206 Suffixes <--- F2* 16 F3*Letter.Series 0.7272957 Letter.Series <--- F3* 17 F3*Pedigrees 0.2468418 Pedigrees <--- F3* 18 F3*Letter.Group 0.4091494 Letter.Group <--- F3* 19 e1 0.1723633 Sentences <--> Sentences 20 e2 0.1698418 Vocabulary <--> Vocabulary 21 e3 0.2684748 Sent.Completion <--> Sent.Completion 22 e4 0.2528108 First.Letters <--> First.Letters 23 e5 0.3873510 4.Letter.Words <--> 4.Letter.Words 24 e6 0.5175675 Suffixes <--> Suffixes 25 e7 0.1496710 Letter.Series <--> Letter.Series 26 e8 0.5003859 Pedigrees <--> Pedigrees 27 e9 0.5517473 Letter.Group <--> Letter.Group 28 1.0000000 F1* <--> F1* 29 1.0000000 F2* <--> F2* 30 1.0000000 F3* <--> F3* 31 1.0000000 g <--> g Compare this solution to the one reported below, and to the sem manual. 4.8 Examining a hierarchical solution A hierarchical solution to this data set was previously found by the omega function (Figure 10). The output of that analysis can be used as a model for a sem analysis. Once again, the stdcoef function helps see the structure. Alternatively, using the omega function on the Thurstone data will create the model for this particular data set. > sem.hi <- sem(om.hi$model,thurstone,213) > summary(sem.hi,digits=2) Model Chisquare = 38.1963 Df = 24 Pr(>Chisq) = 0.03310059 AIC = 80.1963 BIC = -90.47471 Normalized Residuals Min. 1st Qu. Median Mean 3rd Qu. Max. -0.9725000-0.4165000-0.0000001 0.0401000 0.0938600 1.6270000 R-square for Endogenous Variables F1 F2 F3 Sentences Vocabulary 0.6758 0.6112 0.6642 0.8185 0.8351 Sent.Completion First.Letters 4.Letter.Words Suffixes Letter.Series 0.7329 0.6985 0.6355 0.4936 0.6097 Pedigrees Letter.Group 0.5186 0.4949 Parameter Estimates Estimate Std Error z value Pr(> z ) gf1 1.4438115 0.25653564 5.628113 1.821922e-08 gf2 1.2538296 0.21136562 5.932041 2.991910e-09 38

Omega Sentences Vocabulary Sent.Completion 0.91 0.89 0.83 F1 First.Letters 0.86 0.78 4.Letter.Words 0.74 F2 0.76 g Suffixes 0.63 0.37 0.69 Letter.Series Letter.Group Pedigrees 0.21 0.84 0.64 0.47 F3 Figure 14: Hierarchical analysis of the Thurstone 9 variable problem using an exploratory algorithm can provide the appropriate sem code for analysis using the sem package. 39

gf3 1.4065517 0.26890804 5.230605 1.689563e-07 F1Sentences 0.5151232 0.06292248 8.186632 2.686376e-16 F1Vocabulary 0.5203104 0.06338431 8.208820 2.233734e-16 F1Sent.Completion 0.4874316 0.06081528 8.014954 1.101786e-15 F2First.Letters 0.5211221 0.06106205 8.534304 1.410015e-17 F24.Letter.Words 0.4970664 0.05902388 8.421446 3.718664e-17 F2Suffixes 0.4380644 0.05595794 7.828458 4.938915e-15 F3Letter.Series 0.4524352 0.06596903 6.858297 6.968649e-12 F3Pedigrees 0.4172903 0.06215816 6.713363 1.901887e-11 F3Letter.Group 0.4076312 0.06131399 6.648258 2.965820e-11 e1 0.1814979 0.02847741 6.373397 1.848862e-10 e2 0.1649304 0.02776938 5.939292 2.862558e-09 e3 0.2671331 0.03336340 8.006771 1.177597e-15 e4 0.3015024 0.05102191 5.909274 3.436179e-09 e5 0.3645010 0.05263547 6.925008 4.359513e-12 e6 0.5064150 0.05962608 8.493180 2.010593e-17 e7 0.3903313 0.05933649 6.578268 4.759607e-11 e8 0.4813697 0.06224844 7.733041 1.050075e-14 e9 0.5051017 0.06332869 7.975875 1.513055e-15 gf1 F1 <--- g gf2 F2 <--- g gf3 F3 <--- g F1Sentences Sentences <--- F1 F1Vocabulary Vocabulary <--- F1 F1Sent.Completion Sent.Completion <--- F1 F2First.Letters First.Letters <--- F2 F24.Letter.Words 4.Letter.Words <--- F2 F2Suffixes Suffixes <--- F2 F3Letter.Series Letter.Series <--- F3 F3Pedigrees Pedigrees <--- F3 F3Letter.Group Letter.Group <--- F3 e1 Sentences <--> Sentences e2 Vocabulary <--> Vocabulary e3 Sent.Completion <--> Sent.Completion e4 First.Letters <--> First.Letters e5 4.Letter.Words <--> 4.Letter.Words e6 Suffixes <--> Suffixes e7 Letter.Series <--> Letter.Series e8 Pedigrees <--> Pedigrees e9 Letter.Group <--> Letter.Group Iterations = 54 > stdcoef(sem.hi,digits=2) Std. Estimate 1 gf1 0.8220754 F1 <--- g 2 gf2 0.7817998 F2 <--- g 3 gf3 0.8150140 F3 <--- g 4 F1Sentences 0.9047111 Sentences <--- F1 5 F1Vocabulary 0.9138214 Vocabulary <--- F1 6 F1Sent.Completion 0.8560764 Sent.Completion <--- F1 7 F2First.Letters 0.8357617 First.Letters <--- F2 8 F24.Letter.Words 0.7971819 4.Letter.Words <--- F2 9 F2Suffixes 0.7025560 Suffixes <--- F2 10 F3Letter.Series 0.7808129 Letter.Series <--- F3 11 F3Pedigrees 0.7201599 Pedigrees <--- F3 40

12 F3Letter.Group 0.7034902 Letter.Group <--- F3 13 e1 0.1814979 Sentences <--> Sentences 14 e2 0.1649304 Vocabulary <--> Vocabulary 15 e3 0.2671331 Sent.Completion <--> Sent.Completion 16 e4 0.3015024 First.Letters <--> First.Letters 17 e5 0.3645010 4.Letter.Words <--> 4.Letter.Words 18 e6 0.5064151 Suffixes <--> Suffixes 19 e7 0.3903313 Letter.Series <--> Letter.Series 20 e8 0.4813697 Pedigrees <--> Pedigrees 21 e9 0.5051016 Letter.Group <--> Letter.Group 22 0.3241920 F1 <--> F1 23 0.3887891 F2 <--> F2 24 0.3357521 F3 <--> F3 25 1.0000000 g <--> g > anova(sem.hi,sem.bi) LR Test for Difference Between Models Model Df Model Chisq Df LR Chisq Pr(>Chisq) sem.hi 24 38.196 sem.bi 18 24.216 6 13.98 0.02986 * --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Using the Thurstone data set, we see what happens when a hierarchical model is applied to real data. The exploratory structure derived from the omega function (Figure 14) provides estimates in close approximation to those found using sem. The model definition created by using omega is the same hierarchical model discussed in the sem help page. The bifactor model, with 6 more parameters does provide a better fit to the data than the hierarchical model. Similar analyses can be done with other data that are organized hierarchically. Examples of these analyses are analyzing the 14 variables of holzinger and the 16 variables of reise. The output from the following analyses has been limited to just the comparison between the bifactor and hierarchical solutions. > om.holz.bi <- omega(holzinger,4) > sem.holz.bi <- sem(om.holz.bi$model,holzinger,355) > om.holz.hi <- omega(holzinger,4,sl=false) > sem.holz.hi <- sem(om.holz.hi$model,holzinger,355) > anova(sem.holz.bi,sem.holz.hi) LR Test for Difference Between Models Model Df Model Chisq Df LR Chisq Pr(>Chisq) sem.holz.bi 63 147.66 sem.holz.hi 73 178.79 10 31.129 0.0005587 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 41

4.9 Estimating Omega using EFA followed by CFA The function omegasem combines both an exploratory factor analysis using omega, then calls the appropriate sem functions and organizes the results as in a standard omega analysis. An example is found from the Thurstone data set of 9 cognitive variables: > om.sem <- omegasem(thurstone,n.obs=213) Call: omegasem(m = Thurstone, n.obs = 213) Omega Call: omega(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip, digits = digits, title = title, sl = sl, labels = labels, plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option) Alpha: 0.89 G.6: 0.91 Omega Hierarchical: 0.74 Omega H asymptotic: 0.79 Omega Total 0.93 Schmid Leiman Factor loadings greater than 0.2 g F1* F2* F3* h2 u2 p2 Sentences 0.71 0.57 0.82 0.18 0.61 Vocabulary 0.73 0.55 0.84 0.16 0.63 Sent.Completion 0.68 0.52 0.73 0.27 0.63 First.Letters 0.65 0.56 0.73 0.27 0.57 4.Letter.Words 0.62 0.49 0.63 0.37 0.61 Suffixes 0.56 0.41 0.50 0.50 0.63 Letter.Series 0.59 0.61 0.72 0.28 0.48 Pedigrees 0.58 0.23 0.34 0.50 0.50 0.66 Letter.Group 0.54 0.46 0.53 0.47 0.56 With eigenvalues of: g F1* F2* F3* 3.58 0.96 0.74 0.71 general/max 3.71 max/min = 1.35 mean percent general = 0.6 with sd = 0.05 and cv of 0.09 Explained Common Variance of the general factor = 0.6 The degrees of freedom are 12 and the fit is 0.01 The number of observations was 213 with Chi Square = 2.82 with prob < 1 The root mean square of the residuals is 0.01 The df corrected root mean square of the residuals is 0.01 RMSEA index = 0 and the 90 % confidence intervals are NA NA BIC = -61.51 Compare this with the adequacy of just a general factor and no group factors The degrees of freedom for just the general factor are 27 and the fit is 1.48 The number of observations was 213 with Chi Square = 307.1 with prob < 2.8e-49 The root mean square of the residuals is 0.14 The df corrected root mean square of the residuals is 0.16 RMSEA index = 0.224 and the 90 % confidence intervals are 0.199 0.243 BIC = 162.35 42

Measures of factor score adequacy g F1* F2* F3* Correlation of scores with factors 0.86 0.73 0.72 0.75 Multiple R square of scores with factors 0.74 0.54 0.52 0.56 Minimum correlation of factor score estimates 0.49 0.08 0.03 0.11 Total, General and Subset omega for each subset g F1* F2* F3* Omega total for total scores and subscales 0.93 0.92 0.83 0.79 Omega general for total scores and subscales 0.74 0.58 0.50 0.47 Omega group for total scores and subscales 0.16 0.35 0.32 0.32 Omega Hierarchical from a confirmatory model using sem = 0.79 Omega Total from a confirmatory model using sem = 0.93 With loadings of g F1* F2* F3* h2 u2 Sentences 0.77 0.49 0.83 0.17 Vocabulary 0.79 0.45 0.83 0.17 Sent.Completion 0.75 0.40 0.73 0.27 First.Letters 0.61 0.61 0.75 0.25 4.Letter.Words 0.60 0.51 0.61 0.39 Suffixes 0.57 0.39 0.48 0.52 Letter.Series 0.57 0.73 0.85 0.15 Pedigrees 0.66 0.25 0.50 0.50 Letter.Group 0.53 0.41 0.45 0.55 With eigenvalues of: g F1* F2* F3* 3.88 0.61 0.79 0.76 Comparing the two models graphically (Figure 15 with Figure 13 shows that while not identical, they are very similar. The sem version is basically a forced simple structure. Notice that the values of ω h are not identical from the EFA and CFA models. The CFA solution yields higher values of ω h because, by forcing a pure cluster solution (no cross loadings), the correlations between the factors is forced to be through the g factor. 5 Summary and conclusion The use of exploratory and confirmatory models for understanding real data structures is an important advance in psychological research. To understand these approaches it is helpful to try them first on baby data sets. To the extent that the models we use can be tested on simple, artificial examples, it is perhaps easier to practice their application. The psych tools for simulating structural models and for specifying models are a useful supplement to the power of packages such as sem. The techniques that can be used on simulated data set can also be applied to real data sets. 43

Omega from SEM Sentences 0.8 0.8 Vocabulary Sent.Completion 0.5 0.5 0.4 F1* 0.8 0.6 First.Letters 0.6 g 0.6 4.Letter.Words 0.5 F2* 0.6 Suffixes 0.4 0.6 0.7 0.5 Letter.Series Pedigrees 0.7 0.2 0.4 F3* Letter.Group Figure 15: Confirmatory Omega structure using omegasem 44

> sessioninfo() R Under development (unstable) (2015-07-04 r68627) Platform: x86_64-apple-darwin13.4.0 (64-bit) Running under: OS X 10.10.4 (Yosemite) locale: [1] C attached base packages: [1] stats graphics grdevices utils datasets methods base other attached packages: [1] sem_3.1-6 GPArotation_2014.11-1 psych_1.5.6 loaded via a namespace (and not attached): [1] Rcpp_0.11.6 rstudioapi_0.3.1 magrittr_1.5 splines_3.3.0 [5] MASS_7.3-42 mnormt_1.5-3 arm_1.8-5 lattice_0.20-31 [9] mi_1.0 minqa_1.2.4 stringr_1.0.0 tcltk_3.3.0 [13] tools_3.3.0 matrixcalc_1.0-3 parallel_3.3.0 grid_3.3.0 [17] nlme_3.1-121 coda_0.17-1 DiagrammeR_0.7 htmltools_0.2.6 [21] abind_1.4-3 lme4_1.1-8 digest_0.6.8 Matrix_1.2-1 [25] nloptr_1.0.4 htmlwidgets_0.5 stringi_0.5-5 boot_1.3-17 [29] stats4_3.3.0 References Adams, D. (1980). The hitchhiker s guide to the galaxy. Harmony Books, New York, 1st American edition. Bechtoldt, H. (1961). An empirical study of the factor analysis stability hypothesis. Psychometrika, 26(4):405 432. Fox, J. (2006). Structural equation modeling with the sem package in R. Structural Equation Modeling, 13:465 486. Fox, J. (2009). sem: Structural Equation Models. R package version 0.9-15.. Fox, J. Nie, Zhenghua and Byrnes, J. (2013) sem: Structural Equation Models. R package version 3.1-3. Holzinger, K. and Swineford, F. (1937). The bi-factor method. Psychometrika, 2(1):41 54. Jensen, A. R. and Weng, L.-J. (1994). What is a good g? Intelligence, 18(3):231 258. McDonald, R. P. (1999). Test theory: A unified treatment. L. Erlbaum Associates, Mahwah, N.J. Rafaeli, E. and Revelle, W. (2006). A premature consensus: Are happiness and sadness truly opposite affects? Motivation and Emotion, 30(1):1 12. 45

Reise, S., Morizot, J., and Hays, R. (2007). The role of the bifactor model in resolving dimensionality issues in health outcomes measures. Quality of Life Research, 16(0):19 31. Revelle, W. (2014). psych: Procedures for Personality and Psychological Research. Northwestern University, Evanston. R package version 1.4.1 Rosseel, Y. (2012). llavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2):1-36. R package version 0.5-14. Schmid, J. J. and Leiman, J. M. (1957). The development of hierarchical factor solutions. Psychometrika, 22(1):83 90. Thurstone, L. L. and Thurstone, T. G. (1941). Factorial studies of intelligence. The University of Chicago press, Chicago, Ill. 46

Index anova, 29, 30 bifactor, 7, 8, 23, 41 circumplex structure, 8 cluster.cor, 3 congeneric, 5 describe, 3 fa, 14, 32 factanal, 14, 32 factor.congruence, 20 factor.pa, 3, 20 guttman, 3 hierarchical, 8 holzinger, 23, 41 ICLUST, 3, 27 lavaan, 3 minimum residual, 32 oblimin, 22, 32 omega, 3, 8, 23, 35, 38, 41, 42 omega.graph, 28 omegasem, 42, 44 pairs.panels, 3 phi.list, 13, 14, 28 principal, 3, 19, 20 principal components, 19 Promax, 14, 19, 22 promax, 22 psych, 3, 22, 28, 43 quartimin, 19 R function anova, 29, 30 bifactor, 7, 23 cluster.cor, 3 describe, 3 fa, 14, 32 factanal, 14, 32 factor.congruence, 20 factor.pa, 3, 20 guttman, 3 holzinger, 23, 41 ICLUST, 3, 27 oblimin, 22, 32 omega, 3, 8, 23, 35, 38, 41, 42 omega.graph, 28 omegasem, 42, 44 pairs.panels, 3 phi.list, 13, 14, 28 principal, 3, 19, 20 Promax, 14, 19, 22 promax, 22 psych, 3 psych package bifactor, 7, 23 cluster.cor, 3 describe, 3 fa, 14, 32 factor.congruence, 20 factor.pa, 3, 20 guttman, 3 holzinger, 23, 41 ICLUST, 3, 27 oblimin, 32 omega, 3, 8, 23, 35, 38, 41, 42 omega.graph, 28 omegasem, 42, 44 pairs.panels, 3 phi.list, 13, 14, 28 principal, 3, 19, 20 47

Promax, 14, 19, 22 psych, 3 reise, 23, 41 score.items, 3 sim, 4 sim.circ, 8 sim.congeneric, 5, 7 sim.hierarchical, 7 sim.hierarcical, 4 sim.item, 8 sim.structure, 8, 16 structure.diagram, 16, 28 structure.graph, 28, 34 structure.list, 13, 14, 28 structure.sem, 28 Thurstone, 35, 38 VSS, 3 quartimin, 19 reise, 23, 41 Rgraphviz, 28 score.items, 3 sem, 23, 28, 35, 38, 41, 42 sim, 4 sim.circ, 8 sim.congeneric, 5, 7 sim.hierarchical, 7 sim.hierarcical, 4 sim.item, 8 sim.structure, 8, 16 stdcoef, 35, 38 structure.diagram, 16, 28 structure.graph, 28, 34 structure.list, 13, 14, 28 structure.sem, 28 summary, 35 Thurstone, 35, 38 varimax, 19 VSS, 3 R package lavaan, 3 psych, 3, 22, 28, 43 Rgraphviz, 3 sem, 3, 28, 35, 38, 43 reise, 23, 41 Rgraphviz, 3, 28 rotated, 19 score.items, 3 sem, 3, 23, 28, 35, 38, 41 43 sim, 4 sim.circ, 8 sim.congeneric, 5, 7 sim.hierarchical, 7 sim.hierarcical, 4 sim.item, 8 sim.structure, 8, 16 simple structure, 8, 19 stdcoef, 35, 38 structure.diagram, 16, 28 structure.graph, 28, 34 structure.list, 13, 14, 28 structure.sem, 28 summary, 35 tau, 5 Thurstone, 35, 38 varimax, 19 VSS, 3 48